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The questions from paper 27 of the mathematical tripos part iii exam, focusing on morse theory. Topics include showing that a continuous map from the n-sphere to a cell complex is homotopic to a map into the n-skeleton, stating and proving the morse lemma, defining morse functions, gradient-like vector fields, and the morse-smale transversality condition, and proving properties of morse functions on surfaces and the grassmannian.
Typology: Exams
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Friday 6 June 2003 1.30 to 4.
Attempt FOUR questions.
There are six questions in total. The questions carry equal weight.
1 Let X be a cell complex, let Xn ⊆ X denote the n-skeleton (the union of cells of dimension no more than n) and f : Sn^ → X a continuous map from the n-sphere to X. Show carefully that f is homotopic to a continuous map g : Sn^ → Xn. You should clearly state any theorems that you use in the process.
2 State and prove the Morse Lemma.
3 Define the notions of Morse function, gradient-like vector field, and the Morse-Smale transversality condition. Explain (without detailed proof) how the latter allows you to compute the integral homology of a manifold, and illustrate it for the height function on the Klein bottle.
4 Let Σg be the closed orientable surface of genus g. Prove that any Morse function on S^1 × Σg must have at least 4g + 4 critical points, and describe a Morse function which achieves this minimum.
5 Recall that the Grassmannian Gr(k, n) of complex k-dimensional subspaces of Cn is identifiable with the manifold of rank k projection matrices of size n × n. Choose a diagonal matrix C = diag(c 1 ,... , cn) with c 1 <... < cn, and consider the function P 7 → f (P ) = T r(CP ). Determine the critical points and their indices, prove that this is a Morse function and hence describe the homology of Gr(k, n). If it helps, you may specialize to concrete values, such as Gr(2, 4), but you may not choose k = 0, 1 , n − 1 , n.
[You may assume that Gr(k, n) is a submanifold of the space of complex n × n matrices, and that its tangent space at P is spanned by the infinitesimal conjugation action of unitary matrices.]
6 Explain what is meant by attaching a handle, with reference to framed links in R^3 , and explain how you can construct the four-manifold S^2 × S^2 in this manner.
Paper 27
